# Bridging the Physics-Data Gap with FNO-Guided Conditional Flow Matching: Designing Inductive Bias through Hierarchical Physical Constraints

arXiv: https://arxiv.org/abs/2510.08295 · HTML: https://mlsanity.com/paper/2510.08295/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 17 of 17)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs](https://mlsanity.com/paper/1706.02633/) (2017)
- [TimeVAE: A Variational Auto-Encoder for Multivariate Time Series Generation](https://mlsanity.com/paper/2111.08095/) (2021)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [A physics-informed GAN Framework based on Model-free Data-Driven Computational Mechanics](https://mlsanity.com/paper/2310.20308/) (2023)

## Abstract (arXiv)

Conventional time-series generation often ignores domain-specific physical constraints, limiting statistical and physical consistency. We propose a hierarchical framework that embeds the inherent hierarchy of physical laws-conservation, dynamics, boundary, and empirical relations-directly into deep generative models, introducing a new paradigm of physics-informed inductive bias. Our method combines Fourier Neural Operators (FNOs) for learning physical operators with Conditional Flow Matching (CFM) for probabilistic generation, integrated via time-dependent hierarchical constraints and FNO-guided corrections. Experiments on harmonic oscillators, human activity recognition, and lithium-ion battery degradation show 16.3% higher generation quality, 46% fewer physics violations, and 18.5% improved predictive accuracy over baselines.